a characterization of the radical of an ideal

Proposition 1.

Proof.

Suppose x∈I, and P is a prime ideal containing I. Then R-P is an m-system (http://planetmath.org/MSystem). If x∈R-P, then (R-P)∩I≠∅, contradicting the assumption that I⊆P. Therefore x∉R-P. In other words, x∈P, and we have one of the inclusions.

Conversely, suppose x∉I. Then there is an m-system S containing x such that S∩I=∅. Enlarge I to a prime ideal Pdisjoint from S, so that x∉P (we can do this; for a proof, see the second remark in this entry (http://planetmath.org/MSystem)). By contrapositivity, we have the other inclusion.
∎

Remark. This shows that every prime ideal is a radical ideal: for P is the intersection of all prime ideals containing P, and if P is itself prime, then P=P.